Evolution of highly symmetric curves under the shrinking curvature flow

Abstract

Under the shrinking curvature flow with inner normal velocity V = kα(α > 1), it is shown that highly symmetric, locally convex initial curves evolve into a point asymptotically like an multi‐circles. The proof relies on a crucial use of Bonnensen inequality for highly symmetric, locally convex curves. Copyright © 2016 John Wiley & Sons, Ltd.

On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

“…We finally mention that, after Abresch and Langer's celebrated study [1], the behavior of multiply winding rotationally symmetric curves is investigated by several authors for the curve shortening flow [3,20,24,25,44,45] and other second order geometric flows [13,[46][47][48]; in particular, a kind of stability result for multiply covered circles is obtained by Wang [45], in the same spirit of our result. We remark that all these studies focus on locally convex curves, thus using the property of second order.…”

On the isoperimetric inequality and surface diffusion flow for multiply winding curves

“…We finally mention that, after Abresch and Langer's celebrated study [1], the behavior of multiply winding rotationally symmetric curves is investigated by several authors for the curve shortening flow [25,24,3,44,45,20] and other second order geometric flows [47,48,13,46]; in particular, a kind of stability result for multiply covered circles is obtained by Wang [45], in the same spirit of our result. We remark that all these studies focus on locally convex curves, thus using the property of second order.…”

Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow

“…This is a very special phenomenon in the planar geometry, because compact and locally convex hypersurfaces in higher dimensional Euclidean spaces are all convex ones (see Hadamard's theorem [22] or [23]). Due to this reason, the curvature flows of locally convex curves in the plane arose some particular interests in the past several years (see Chen-Wang-Yang [6], Wang-Li-Chao [29], Wang-Wo-Yang [30]). Locally convex curves also play an important role in understanding the asymptotic behavior of the famous curve shortening flow (see Abresch-Langer [1], Altschuler [2], Angenent [4] and Epstein-Gage [10]) and its generalization (see Andrews [3]).…”